A crime is an offence that merits community condemnation and punishment, usually by way of fine or imprisonment. This is different from a civil wrong (a tort), which is an action against an individual that requires compensation or restitution.
Criminal offences are normally prosecuted by the State or the Commonwealth, whereas it is usually up to an individual to take a civil action to court. It is also possible for an individual to begin criminal proceedings, but this is very rare.
Some matters, such as assault, can be both crimes and civil wrongs at the same time. The police can prosecute for assault and the victim can take civil action to recover money (or some other kind of compensation) for any injury suffered.
It is not always easy to tell when something is a crime. A person who takes money without permission commits a criminal offence, whereas a person who fails to pay back money commits a civil wrong (not a crime). Although a civil action can be commenced to recover the money, the borrower can only be prosecuted for a criminal offence if fraud is involved.
Whether or not the police decide to charge a wrongdoer with a criminal offence is entirely their decision. A victim of crime cannot force the police to prosecute an offender but it is possible, although not common, to make a private prosecution. It is advisable to get legal advice if you are considering this.
There are a range of sources of law which establish the existence of crimes.
Some crimes exist under Commonwealth Acts, such as the Crimes Act 1914 (Cth) and the Customs Act 1901 (Cth). Some crimes exist only at common law (judge made law, not found in legislation). Most criminal offences have been codified (put into legislation) but some common law criminal offences still exist in jurisdictions such as South Australia.
Most criminal offences in South Australia are found in the Criminal Law Consolidation Act 1935 (SA) and the Summary Offences Act 1953 (SA), as well as the Controlled Substances Act 1984 (SA) and various traffic legislation.